Quantiﬁers extend the expressive power of the logical notation
but can be motivated as abbreviations.

Existential quantiﬁer. This is denoted by ∃ and is read as “there
exists a...”

∃I∈{7, 8, 9}IsPrime(I)

This quantiﬁed expression can be read as:

there exists a value in the set {7, 8, 9} which satisﬁes thetruth-valued function IsPrime.

The I is the bound identiﬁer, the I ∈ {…} is the constraint
and the expression after the dot is the body of the quantiﬁed
expression.

Any free occurrences of the bound identiﬁer within the body
become bound in the quantiﬁed expression. All such occurrences
refer to the bound identiﬁer. The quantiﬁers thus provide a way
of deﬁning a context for free identiﬁers.

The expression ∃I ∈{11, 12, 13}IsOdd(I)

is indeed true as it is equivalent to the proposition:

IsOdd(11)∨IsOdd(12)∨IsOdd(13)

The reason that quantiﬁers extend the expressive power of the
logic is that the sets in the constraint of a quantiﬁed expression
can be inﬁnite. Such an expression abbreviate a disjunction which
could never be completely written. For example:

∃I ∈ ℕ_{1}IsPrime(I)

or:

∃I ∈ ℕ_{1}¬IsPrime(2^{I}− 1)

express facts about prime numbers.

Universal Quantiﬁers. Denoted by ∀ and is read as “for all...”.

Just as a disjunction can be viewed as an existentially quantiﬁed
expression, a conjunction such as:

IsEven(2)∧IsEven(4)∧IsEven(8)

can be written as a universally quantiﬁed expression:

∀I ∈{2, 4, 8}IsEven(I)

here again, universal quantiﬁcation increase the expressive power
of the logic when we deal with inﬁnite sets: